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Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University 16 January 2013 Mathematics Education Colloquium, Michigan State University
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Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

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Page 1: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of AlgebraRose Mary Zbiek, The Pennsylvania State University16 January 2013Mathematics Education Colloquium, Michigan State University

Page 2: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Agenda

Biases and Background

Literature Landscape

Questions and Cluster

Page 3: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Research and Teaching

Research

PracticePracticeResearc

hResearc

hPractice

Research Practice

Page 4: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Basic BiasesUsing Literature

Laundry lists are inadequate.

Summaries are helpful.

Syntheses are useful.

Digging more deeply into the literature and probing beyond the surface is necessary.

Using Research

Findings

Tasks

Theories

Constructs

Page 5: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Technology?Mathematics Technology

Mathematics technology, such as

Calculators

Computer algebra systems (CAS)

Dynamic geometry

Dynamic statistics

Spreadsheets

Applets

Manipulatives

Note: Content and tool lines are blurred.

Other Technology

Communication collaboration technology, such as

Online learning venues

Pedagogical tools, such as

Intelligent tutors

New forms, such as

Classroom collaboration tools

Tablets

Page 6: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Literature LandscapeMotivations

Develop technology products

Create technology-present curriculum

Fascination with the technology

“Prove” how great technology is

Foci

Learning, Teaching

Students, Teachers

Performance, Achievement

Strategy, Problem solving

Affect, Motivation, Identity

Policy

Page 7: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Literature LandscapeApproach to Algebra

Function

(blur with calculus)

Structure

(blur with skills)

Problem solving

Modeling

Boundaries can be blurred.

Missing Mountains

Beginning algebra or Algebra I

Recently developed technologies

Theoretical grounding (more than a taxonomy)

We do have vital valleys.

Page 8: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Learning about Functions?

Dominance of graphing calculators in particular and graphing utilities in general

Dominant CAS use as graphing tool

Multiple representations as major issue

Heid & Blume, 2008Yerushalmy, 2006Heid, Thomas, &

Zbiek, IP

Page 9: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Rule of Four

Graphs

Numbers

Words

Symbols

Page 10: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Rule of Four ±k

Graphs

Numbers

Words

Symbols

Symbols

Symbols

Symbols

Symbols

Graphs

Page 11: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Example: Two Graph Registers

Cartesian

Dynamap/Dynagraph

Page 12: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Rule of Four ±k, Now Linked

Graphs

Numbers

Words

Symbols

Symbols

Symbols

Symbols

Symbols

Graphs

Note that technology does not (yet? typically?) connect the words or situation to the other representations.

Page 13: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Example: Lots of Links

Parameter/Slider

Bidirectional Link

Page 14: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Students with Strong Skills?

… influenced by their confidence in their own algebraic skills, [they are] less likely to consider a machine-generated graph as authoritatively correct.

Were more reluctant to use graphical representations even when the representations provided much more accessible information than a symbolic formula when using a traditional curriculum without technology.

Hart, 1991Zbiek et al., 2007, p.

1176

Page 15: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Struggling Learners?

Approach to toolsTend to use tool to replace tedious computations

Leverage tool use to skip something that they could not do on their own

Regarding representationsDelay using symbols

Focus on numerical and graphical methods

Yerushalmy, 2006

Page 16: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Privileging Representations?

Two teachers intensely plan a (calculus) lesson together, including CAS use.

Their enacted lessons look very different.

Their students score similarly in total but use different strategies and different representations.

Offered explanation: The teachers privilege different representational forms.

Kendal & Stacey, 2002

Page 17: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

And Preservice Teachers?

Functions can be useful models. How do preservice teachers develop function models?

Four ways of developing a model with function-fitting tool and graphing utility:

Technology determines the model—R2 rules.

Choose one of the technology-generated models based on characteristics of families of functions.

Choose function family based on characteristics and then use technology to create and revise symbolic from to match data.

Go for a known or assumed relationship (e.g., “looks like doubling” for the first two data points).

Zbiek, 1998

Page 18: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Points about Multiple Reps

Inhibited by symbolically driven environments

Mathematical feedback in one register might inform work in another register or cause a change in solution path. In particular, mathematical feedback seems important when the goal is to produce symbolic expressions or implement substantial procedures.

Examples of mathematical feedback include match of graph and data, value of R2, and unexpected extrema.

Assumes one is looking for connections or links between elements and characteristics of representations.

Hillel, Kieran, & Gurtner, 1989

Page 19: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Learning to Use Technology?

Any technology that we use is another’s artifact. Learners need to make it their tool.

Instrumental genesis is the process of an individual developing a tool-user relationship with the technology. Both the individual’s thinking and the technology can change in the process.

Guin & Trouche, 1999Verillon & Rabardel,

1995

Page 20: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Teacher Using Technology?

Mishra and Koehler’s TPACK might describe the kinds of knowledge needed.

For the teacher, mathematics technology must become both a mathematics tool and a pedagogical tool.

Play, personal math Use, small scale Recommended use with students, larger scale Implementation with classes, and Assessment of the innovations (PURIA) are potential phases through which even experienced teachers pass.

Teachers develop instrumental orchestrations.

Mishra & Koehler, 2006

Beaudin & Bowers, 1997

Trouche, 2005Zbiek & Hollebrands,

2008

Page 21: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

What about Symbolic Work?

Let’s acknowledge the importance of knowing at least some by-hand procedures.

Question how technology can be used to help students learn and learn about by-hand procedures

Motivation

Reflection

Page 22: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

The Pet Ward Task

Heid & Zbiek, 1995, p. 652

Page 23: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Student Responses

Heid & Zbiek, 1995, p. 652

Page 24: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

The Representations

This is the first time that

students EVER begged me to teach them

how to combine like

terms!

Page 25: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

CIA Results

Learned by-hand methods faster

Learned by-hand methods nearly as well

Appreciated efficiency

“Changed the winners”

Heid & Zbiek, 1995

Page 26: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

The Reality

Motivation or attitude bump with the start of technology use might disappear.

Example: CAS use in “college algebra” settings

Zbiek, 2003

Page 27: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Developing Skills, Concepts, … or

Techniques?Developing a model required more the machine procedure, even if it was only reasoning about R2.

“technique, the nature of technical capacity that goes beyond rote application of procedures.” (p. 1179, emphasis added)

A technique is a method for carrying out, or the ability to perform, a task. … using a technique involves not only routine work. …complex reasoning is required. (p. 128)Hitt & Kieran, 2009, p.

128Zbiek et al., 2007, p.

1179

Page 28: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Tasks to Develop Techniques

Paper and pencil “solution”

CAS “solution”

Reflection and resolution

???

Guzman, Kieran, & Martinez, 2010

Page 29: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

CAS, Tasks, and Teacher

Results given by the CAS provoked in students … the use of the CAS in the context of the designed task led the students to rethink their techniques and explanations and provoked a theoretical reflection that could explain for them the results given by the CAS.

However, such theoretical reflection was not enough …. These results thus suggest that, in spite of good tasks and the use of CAS, in order for students to more fully understand rational expressions and their simplification, including the relation between polynomial division and factored forms within rational expressions, the importance of teacher intervention is inescapable.

Guzman, Kieran, & Martinez, 2010, p.

1501

Page 30: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Points about Procedures

Technology is good for more than checking answers.

Results that challenge and intrigue students can motivate students to reason about symbolic procedures, including procedures that the students have not yet acquired.

Tasks are carefully chosen and sequenced for use with the technology, and the tasks demand resolution of a mathematical dilemma in the students’ eyes.

Teacher prompts reasoning and reflection to go beyond procedure and into technique.

Page 31: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Connected Technology?That’s all good with individual handhelds and desktops. What happens when mathematics technology blends with communication/collaboration technology?

“Show your work” and “look at our work.”

Within-multi-representations these focus on the interactions of students with the multi-representations supported by the software itself (e.g. TI-Nspire) … produced by a single student or by a small group of students using the same device.

Between-multi-representations these principally focus on the interactions that the instrument (e.g. TI-Navigator) triggers and supports amongst students in the classroom, because of the simultaneous access on the shared screen to the solutions produced by different students for the same task.

Arzarello & Robutti, 2010

Page 32: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

There’s More than Algebra?

Research that does not directly address algebra (or function) can offer useful constructs.

An example:

“Drag” in dynamic geometry environments Dynamic CAS or graphing settings

Page 33: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

“Drag” in Dynamic Geometry

Wandering dragging, a somewhat random type of dragging in which the student’s goal is to search for regularities or interesting results that occur when some object is dragged.

Lieu muet dragging (dragging in which the student tacitly or explicitly maintains some condition), the student’s goal is to preserve some regularity in the drawing.

Dragging test (dragging to test a hypothesis), has a different goal, namely, to determine whether a conjecture (the student’s or someone else’s) is true.

Arzarello et al., 1998

Page 34: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Wandering

Lieu muet

Dragging test

Examples of DGE Dragging

Page 35: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Parallels in “DAE*”

Guin and Trouche (1999) observed in weaker students aged 15 to 16 “avoidance strategies” such as random trials and “zapping” to other commands in the same menu, similar to what Ball and Stacey (2005) reported.

DAE: My suggestion that we need to think about “dynamic algebra environments”

Guin & Trouche, 1999Ball & Stacey, 2005,Zbiek et al., 2007, p.

1185

Page 36: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Wandering

Lieu muet

Dragging test

Examples of DGE Dragging

Page 37: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Help for Learners?

Feedback can alert user to an error but might not help the user to know why the error exists.

Tutoring programs / intelligent tutors

Possibility that pedagogical technology blends with mathematical technology

Koedinger, 1998

Page 38: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

In Summary…All those representations are great but the student, not only the technology, needs to link them. Linking is more than translation; it requires connecting elements and characteristics.

Reasoning and procedure thrive in harmony for students to develop mathematical/statistical models and techniques.

Symbols are a challenge for some students. Symbols in technology settings can be overdone or underprivileged.

Students and teachers develop a relationship to technology.

Mathematical and pedagogical feedback can be great if students use it, and it should be rationale beyond judgment.

Page 39: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Lurking Questions

What are the effects on understanding of function, procedures, modeling, and other things of representations that are fully bi-directionally linked in technology?

How can students capitalize on linked multiple representations and mathematical feedback

To develop robust understanding of symbols?

To develop representational fluency?

To develop techniques?Sandoval, Bell, Coleman,

Enyedy, & Suthers, 2000

Zbiek et al., 2007

Page 40: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

Lurking Question

What about technology environments that blend mathematics technology with other kinds of technology?

Page 41: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

LURKING Question

How much of what by-hand content needs to be learned?

Page 42: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

And the story continues…Thank [email protected]

Research Practice

Page 43: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

ReferencesArzarello, F., & Robutti, O. (2010). Multimodality in multi-representational environments. ZDM (42), 715 – 731.

Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in Cabri and modalities of transition from conjectures to proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 32–39). South Africa: University of Stellenbosch.

Ball, L., & Stacey, K. (2005). Teaching strategies for developing judicious technology use. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments, 2005 Yearbook of the National Council of Teachers of Mathematics (pp. 3–15). Reston, VA: National Council of Teachers of Mathematics.

Beaudin, M. & Bowers, D. (1997). Logistics for facilitating CAS instruction. In J. Berry, J. Monaghan, M. Kronfellner, & B. Kutzler (Eds.), The state of computer algebra in mathematics education (pp. 126–135). Lancashire, England: Chartwell-York.

Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3, 195–227

Page 44: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

References (continued)Guzmán, J., Kieran, C., & Martinez, C. (2010). The role of computer algebra systems (cas) and a task on the simplification of rational expressions designed with a technical-theoretical approach. In Brosnan, P., Erchick, D. B., & Flevares, L. (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1497-1505). Columbus, OH: The Ohio State University.

Hart, D. (1991). Building concept images: Supercalculators and students’ use of multiple representations in calculus (Doctoral dissertation, Oregon State University, 1991). Dissertation Abstracts International 52(12), 4254.

Heid, M. K., & Blume, G. W. (2008). Technology and the development of algebraic understanding. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Syntheses, cases, and perspectives. Vol. 1: Research syntheses (pp. 55-108). Charlotte, NC: Information Age Publishing.

Heid, M. K., Thomas, M., & Zbiek, R. M. (in press). How might computer algebra systems change the role of algebra in the school curriculum? In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.) Third International Handbook of Mathematics Education. Dordrecht, The Netherlands: Springer.

Page 45: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

References (continued)Heid, M. K., & Zbiek, R. M. (1995). A technology-intensive approach to algebra. Mathematics Teacher, 88(8), 650-656.

Hillel, J., Kieran, C., & Gurtner, J. (1989). Solving structured geometric tasks on the computer: The role of feedback in generating strategies. Educational Studies in Mathematics, 20, 1-39

Hitt, F., & Kieran, C. (2009). Constructing knowledge via a peer interaction in a CAS environment with tasks designed from a Task-Technique-Theory perspective. International Journal of Computers for Mathematical Learning, 14, 121-152. (available from Springer On-line)

Kendal, M., & Stacey, K. (2002). The impact of teacher privileging on learning differentiation with technology. International Journal of Computers for Mathematical Learning, 6(2), 143-265.

Koedinger, K. R. (1998). Conjecturing and argumentation in high-school geometry students. In Lehrer, R., & Chazan, D. (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 319-348). Mahwah, NJ: Lawrence Erlbaum.

Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A new framework for teacher knowledge . Teachers College Record. 108 (6), 1017-1054.

Page 46: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

References (continued)Sandoval, W. A., Bell, P., Coleman, E., Enyedy, N., & Suthers, D. (2000, April). Designing knowledge representations for epistemic practices in science learning. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.

Trouche, L. (2005). Instrumental genesis, individual and social aspects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators (pp. 197-230). New York: Springer.

Verillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of though[t] in relation to instrumented activity. European Journal of Psychology in Education 9, 77–101.

Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving algebra word problems with graphing software. Journal for Research in Mathematics Education, 37, 356-387.

Zbiek, R. M. (1998). Prospective teachers' use of computing tools to develop and validate functions as mathematical models. Journal for Research in Mathematics Education, 29(2), 184-201.

Page 47: Putting Research on Technology in Mathematics Education to Work to Inform Teaching and Learning of Algebra Rose Mary Zbiek, The Pennsylvania State University.

References (continued)Zbiek, R. M. (2003). Using research to inform teaching and learning with computer algebra systems. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.) Computer algebra systems in mathematics education (pp. 197-216). Reston, VA: National Council of Teachers of Mathematics.

Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. (2007). Research on technology in mathematics education: A perspective of constructs. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1169-1207). Charlotte, NC: Information Age.

Zbiek, R. M., & Hollebrands, K. (2008). A research-informed view of the process of incorporating mathematics technology into classroom practice by inservice and prospective teachers. In M. K. Heid and G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Volume 1 (pp. 287-344). Charlotte, NC: Information Age.